A wide range of applications in science and engineering involve a PDE model
in a domain with perforations, such as perforated metals or air filters.
Solving such perforated domain problems suffers from computational challenges
related to resolving the scale imposed by the geometries of perforations. We
propose a neural network-based mesh-free approach for perforated domain
problems. The method is robust and efficient in capturing various configuration
scales, including the averaged macroscopic behavior of the solution that
involves a multiscale nature induced by small perforations. The new approach
incorporates the derivative-free loss method that uses a stochastic
representation or the Feynman-Kac formulation. In particular, we implement the
Neumann boundary condition for the derivative-free loss method to handle the
interface between the domain and perforations. A suite of stringent numerical
tests is provided to support the proposed method's efficacy in handling various
perforation scales.

提出了一种基于神经网络的无网格方法来解决具有孔洞领域的 PDE 模型问题，该方法能够捕捉各种尺度的配置，包括由小孔洞引起的多尺度特性，并采用了无导数损失方法处理领域与孔洞之间的界面。通过一系列严格的数值测试支持了该方法在处理不同孔洞尺度问题上的有效性。

微孔域中椭圆问题的随机方法

Stochastic approach for elliptic problems in perforated domains

This study analyzes the derivative-free loss method to solve a certain class
of elliptic PDEs using neural networks. The derivative-free loss method uses
the Feynman-Kac formulation, incorporating stochastic walkers and their
corresponding average values. We investigate the effect of the time interval
related to the Feynman-Kac formulation and the walker size in the context of
computational efficiency, trainability, and sampling errors. Our analysis shows
that the training loss bias is proportional to the time interval and the
spatial gradient of the neural network while inversely proportional to the
walker size. We also show that the time interval must be sufficiently long to
train the network. These analytic results tell that we can choose the walker
size as small as possible based on the optimal lower bound of the time
interval. We also provide numerical tests supporting our analysis.

该研究分析了使用神经网络解决一类椭圆型偏微分方程的无导数损失方法。该方法利用费曼 - 卡克公式，结合随机游走和其相应的平均值。我们研究了费曼 - 卡克公式中与时间间隔相关的效果，以及与计算效率、可训练性和采样误差的关系。我们的分析表明，训练损失的偏差与时间间隔和神经网络的空间梯度成正比，与游走大小成反比。我们还表明，时间间隔必须足够长以训练网络。这些分析结果告诉我们可以根据时间间隔的最优下限尽可能选择较小的游走大小。我们还提供数值测试来支持我们的分析。